arXiv subjects

Elchanan Mossel

Publications and source records attributed to Elchanan Mossel.

At least 127 records · Page 7Linked to original sources

arXiv2017

Coalescent-based species tree estimation: a stochastic Farris transform

The reconstruction of a species phylogeny from genomic data faces two significant hurdles: 1) the trees describing the evolution of each individual gene--i.e., the gene trees--may differ from the species phylogeny and 2) the molecular sequences corresponding to each gene often provide limited information about the gene trees themselves. In this paper we consider an approach to species tree reconstruction that addresses both these hurdles. Specifically, we propose an algorithm for phylogeny reconstruction under the multispecies coalescent model with a standard model of site substitution. The multispecies coalescent is commonly used to model gene tree discordance due to incomplete lineage sorting, a well-studied population-genetic effect. In previous work, an information-theoretic trade-off was derived in this context between the number of loci, $m$, needed for an accurate reconstruction and the length of the locus sequences, $k$. It was shown that to reconstruct an internal branch of length $f$, one needs $m$ to be of the order of $1/[f^{2} \sqrt{k}]$. That previous result was obtained under the molecular clock assumption, i.e., under the assumption that mutation rates (as well as population sizes) are constant across the species phylogeny. Here we generalize this result beyond the restrictive molecular clock assumption, and obtain a new reconstruction algorithm that has the same data requirement (up to log factors). Our main contribution is a novel reduction to the molecular clock case under the multispecies coalescent. As a corollary, we also obtain a new identifiability result of independent interest: for any species tree with $n \geq 3$ species, the rooted species tree can be identified from the distribution of its unrooted weighted gene trees even in the absence of a molecular clock.

cs.LG↗

arXiv2017

Opinion Exchange Dynamics

We survey a range of models of opinion exchange. From the introduction: "The exchange of opinions between individuals is a fundamental social interaction... Moreover, many models in this field are an excellent playground for mathematicians, especially those working in probability, algorithms and combinatorics. The goal of this survey is to introduce such models to mathematicians, and especially to those working in discrete mathematics, information theory, optimization, probability and statistics."

math.PR↗

arXiv2017

Noise Stability is computable and low dimensional

Questions of noise stability play an important role in hardness of approximation in computer science as well as in the theory of voting. In many applications, the goal is to find an optimizer of noise stability among all possible partitions of $\mathbb{R}^n$ for $n \geq 1$ to $k$ parts with given Gaussian measures $μ_1,\ldots,μ_k$. We call a partition $ε$-optimal, if its noise stability is optimal up to an additive $ε$. In this paper, we give an explicit, computable function $n(ε)$ such that an $ε$-optimal partition exists in $\mathbb{R}^{n(ε)}$. This result has implications for the computability of certain problems in non-interactive simulation, which are addressed in a subsequent work.

math.PR↗

arXiv2017

Non interactive simulation of correlated distributions is decidable

A basic problem in information theory is the following: Let $\mathbf{P} = (\mathbf{X}, \mathbf{Y})$ be an arbitrary distribution where the marginals $\mathbf{X}$ and $\mathbf{Y}$ are (potentially) correlated. Let Alice and Bob be two players where Alice gets samples $\{x_i\}_{i \ge 1}$ and Bob gets samples $\{y_i\}_{i \ge 1}$ and for all $i$, $(x_i, y_i) \sim \mathbf{P}$. What joint distributions $\mathbf{Q}$ can be simulated by Alice and Bob without any interaction? Classical works in information theory by G{á}cs-K{ö}rner and Wyner answer this question when at least one of $\mathbf{P}$ or $\mathbf{Q}$ is the distribution on $\{0,1\} \times \{0,1\}$ where each marginal is unbiased and identical. However, other than this special case, the answer to this question is understood in very few cases. Recently, Ghazi, Kamath and Sudan showed that this problem is decidable for $\mathbf{Q}$ supported on $\{0,1\} \times \{0,1\}$. We extend their result to $\mathbf{Q}$ supported on any finite alphabet. We rely on recent results in Gaussian geometry (by the authors) as well as a new \emph{smoothing argument} inspired by the method of \emph{boosting} from learning theory and potential function arguments from complexity theory and additive combinatorics.

cs.CC↗

arXiv2016

Linear Sketching over $\mathbb F_2$

We initiate a systematic study of linear sketching over $\mathbb F_2$. For a given Boolean function $f \colon \{0,1\}^n \to \{0,1\}$ a randomized $\mathbb F_2$-sketch is a distribution $\mathcal M$ over $d \times n$ matrices with elements over $\mathbb F_2$ such that $\mathcal Mx$ suffices for computing $f(x)$ with high probability. We study a connection between $\mathbb F_2$-sketching and a two-player one-way communication game for the corresponding XOR-function. Our results show that this communication game characterizes $\mathbb F_2$-sketching under the uniform distribution (up to dependence on error). Implications of this result include: 1) a composition theorem for $\mathbb F_2$-sketching complexity of a recursive majority function, 2) a tight relationship between $\mathbb F_2$-sketching complexity and Fourier sparsity, 3) lower bounds for a certain subclass of symmetric functions. We also fully resolve a conjecture of Montanaro and Osborne regarding one-way communication complexity of linear threshold functions by designing an $\mathbb F_2$-sketch of optimal size. Furthermore, we show that (non-uniform) streaming algorithms that have to process random updates over $\mathbb F_2$ can be constructed as $\mathbb F_2$-sketches for the uniform distribution with only a minor loss. In contrast with the previous work of Li, Nguyen and Woodruff (STOC'14) who show an analogous result for linear sketches over integers in the adversarial setting our result doesn't require the stream length to be triply exponential in $n$ and holds for streams of length $\tilde O(n)$ constructed through uniformly random updates. Finally, we state a conjecture that asks whether optimal one-way communication protocols for XOR-functions can be constructed as $\mathbb F_2$-sketches with only a small loss.

cs.DS↗

arXiv2016

Belief propagation, robust reconstruction and optimal recovery of block models

We consider the problem of reconstructing sparse symmetric block models with two blocks and connection probabilities $a/n$ and $b/n$ for inter- and intra-block edge probabilities, respectively. It was recently shown that one can do better than a random guess if and only if $(a-b)^2>2(a+b)$. Using a variant of belief propagation, we give a reconstruction algorithm that is optimal in the sense that if $(a-b)^2>C(a+b)$ for some constant $C$ then our algorithm maximizes the fraction of the nodes labeled correctly. Ours is the only algorithm proven to achieve the optimal fraction of nodes labeled correctly. Along the way, we prove some results of independent interest regarding robust reconstruction for the Ising model on regular and Poisson trees.

math.PR↗

arXiv2016

Efficient Bayesian Learning in Social Networks with Gaussian Estimators

We consider a group of Bayesian agents who try to estimate a state of the world $θ$ through interaction on a social network. Each agent $v$ initially receives a private measurement of $θ$: a number $S_v$ picked from a Gaussian distribution with mean $θ$ and standard deviation one. Then, in each discrete time iteration, each reveals its estimate of $θ$ to its neighbors, and, observing its neighbors' actions, updates its belief using Bayes' Law. This process aggregates information efficiently, in the sense that all the agents converge to the belief that they would have, had they access to all the private measurements. We show that this process is computationally efficient, so that each agent's calculation can be easily carried out. We also show that on any graph the process converges after at most $2N \cdot D$ steps, where $N$ is the number of agents and $D$ is the diameter of the network. Finally, we show that on trees and on distance transitive-graphs the process converges after $D$ steps, and that it preserves privacy, so that agents learn very little about the private signal of most other agents, despite the efficient aggregation of information. Our results extend those in an unpublished manuscript of the first and last authors.

stat.AP↗

arXiv2016

Density Evolution in the Degree-correlated Stochastic Block Model

There is a recent surge of interest in identifying the sharp recovery thresholds for cluster recovery under the stochastic block model. In this paper, we address the more refined question of how many vertices that will be misclassified on average. We consider the binary form of the stochastic block model, where $n$ vertices are partitioned into two clusters with edge probability $a/n$ within the first cluster, $c/n$ within the second cluster, and $b/n$ across clusters. Suppose that as $n \to \infty$, $a= b+ μ\sqrt{ b} $, $c=b+ ν\sqrt{ b} $ for two fixed constants $μ, ν$, and $b \to \infty$ with $b=n^{o(1)}$. When the cluster sizes are balanced and $μ\neq ν$, we show that the minimum fraction of misclassified vertices on average is given by $Q(\sqrt{v^*})$, where $Q(x)$ is the Q-function for standard normal, $v^*$ is the unique fixed point of $v= \frac{(μ-ν)^2}{16} + \frac{ (μ+ν)^2 }{16} \mathbb{E}[ \tanh(v+ \sqrt{v} Z)],$ and $Z$ is standard normal. Moreover, the minimum misclassified fraction on average is attained by a local algorithm, namely belief propagation, in time linear in the number of edges. Our proof techniques are based on connecting the cluster recovery problem to tree reconstruction problems, and analyzing the density evolution of belief propagation on trees with Gaussian approximations.

stat.ML↗

arXiv2016

Shotgun Assembly of Random Jigsaw Puzzles

In a recent work, Mossel and Ross considered the shotgun assembly problem for a random jigsaw puzzle. Their model consists of a puzzle - an $n\times n$ grid, where each vertex is viewed as a center of a piece. They assume that each of the four edges adjacent to a vertex, is assigned one of $q$ colors (corresponding to "jigs", or cut shapes) uniformly at random. Mossel and Ross asked: how large should $q = q(n)$ be so that with high probability the puzzle can be assembled uniquely given the collection of individual tiles? They showed that if $q = ω(n^2)$, then the puzzle can be assembled uniquely with high probability, while if $q = o(n^{2/3})$, then with high probability the puzzle cannot be uniquely assembled. Here we improve the upper bound and show that for any $\eps > 0$, the puzzle can be assembled uniquely with high probability if $q \geq n^{1+\eps}$. The proof uses an algorithm of $n^{Θ(1/\eps)}$ running time.

math.CO↗

arXiv2016

Noise Stability and Correlation with Half Spaces

Benjamini, Kalai and Schramm showed that a monotone function $f : \{-1,1\}^n \to \{-1,1\}$ is noise stable if and only if it is correlated with a half-space (a set of the form $\{x: \langle x, a\rangle \le b\}$). We study noise stability in terms of correlation with half-spaces for general (not necessarily monotone) functions. We show that a function $f: \{-1, 1\}^n \to \{-1, 1\}$ is noise stable if and only if it becomes correlated with a half-space when we modify $f$ by randomly restricting a constant fraction of its coordinates. Looking at random restrictions is necessary: we construct noise stable functions whose correlation with any half-space is $o(1)$. The examples further satisfy that different restrictions are correlated with different half-spaces: for any fixed half-space, the probability that a random restriction is correlated with it goes to zero. We also provide quantitative versions of the above statements, and versions that apply for the Gaussian measure on $\mathbb{R}^n$ instead of the discrete cube. Our work is motivated by questions in learning theory and a recent question of Khot and Moshkovitz.

math.PR↗

arXiv2016

Invariance principle on the slice

We prove an invariance principle for functions on a slice of the Boolean cube, which is the set of all vectors {0,1}^n with Hamming weight k. Our invariance principle shows that a low-degree, low-influence function has similar distributions on the slice, on the entire Boolean cube, and on Gaussian space. Our proof relies on a combination of ideas from analysis and probability, algebra and combinatorics. Our result imply a version of majority is stablest for functions on the slice, a version of Bourgain's tail bound, and a version of the Kindler-Safra theorem. As a corollary of the Kindler-Safra theorem, we prove a stability result of Wilson's theorem for t-intersecting families of sets, improving on a result of Friedgut.

math.PR↗

arXiv2016

Sequence assembly from corrupted shotgun reads

The prevalent technique for DNA sequencing consists of two main steps: shotgun sequencing, where many randomly located fragments, called reads, are extracted from the overall sequence, followed by an assembly algorithm that aims to reconstruct the original sequence. There are many different technologies that generate the reads: widely-used second-generation methods create short reads with low error rates, while emerging third-generation methods create long reads with high error rates. Both error rates and error profiles differ among methods, so reconstruction algorithms are often tailored to specific shotgun sequencing technologies. As these methods change over time, a fundamental question is whether there exist reconstruction algorithms which are robust, i.e., which perform well under a wide range of error distributions. Here we study this question of sequence assembly from corrupted reads. We make no assumption on the types of errors in the reads, but only assume a bound on their magnitude. More precisely, for each read we assume that instead of receiving the true read with no errors, we receive a corrupted read which has edit distance at most $ε$ times the length of the read from the true read. We show that if the reads are long enough and there are sufficiently many of them, then approximate reconstruction is possible: we construct a simple algorithm such that for almost all original sequences the output of the algorithm is a sequence whose edit distance from the original one is at most $O(ε)$ times the length of the original sequence.

q-bio.GN↗

arXiv2015

Coexistence in preferential attachment networks

We introduce a new model of competition on growing networks. This extends the preferential attachment model, with the key property that node choices evolve simultaneously with the network. When a new node joins the network, it chooses neighbours by preferential attachment, and selects its type based on the number of initial neighbours of each type. The model is analysed in detail, and in particular, we determine the possible proportions of the various types in the limit of large networks. An important qualitative feature we find is that, in contrast to many current theoretical models, often several competitors will coexist. This matches empirical observations in many real-world networks.

physics.soc-ph↗

arXiv2015

On the Correlation of Increasing Families

The classical correlation inequality of Harris asserts that any two monotone increasing families on the discrete cube are nonnegatively correlated. In 1996, Talagrand established a lower bound on the correlation in terms of how much the two families depend simultaneously on the same coordinates. Talagrand's method and results inspired a number of important works in combinatorics and probability theory. In this paper we present stronger correlation lower bounds that hold when the increasing families satisfy natural regularity or symmetry conditions. In addition, we present several new classes of examples for which Talagrand's bound is tight. A central tool in the paper is a simple lemma asserting that for monotone events noise decreases correlation. This lemma gives also a very simple derivation of the classical FKG inequality for product measures, and leads to a simplification of part of Talagrand's proof.

math.CO↗

arXiv2015

A Proof Of The Block Model Threshold Conjecture

We study a random graph model named the "block model" in statistics and the "planted partition model" in theoretical computer science. In its simplest form, this is a random graph with two equal-sized clusters, with a between-class edge probability of $q$ and a within-class edge probability of $p$. A striking conjecture of Decelle, Krzkala, Moore and Zdeborová based on deep, non-rigorous ideas from statistical physics, gave a precise prediction for the algorithmic threshold of clustering in the sparse planted partition model. In particular, if $p = a/n$ and $q = b/n$, $s=(a-b)/2$ and $p=(a+b)/2$ then Decelle et al.\ conjectured that it is possible to efficiently cluster in a way correlated with the true partition if $s^2 > p$ and impossible if $s^2 < p$. By comparison, the best-known rigorous result is that of Coja-Oghlan, who showed that clustering is possible if $s^2 > C p \ln p$ for some sufficiently large $C$. In a previous work, we proved that indeed it is information theoretically impossible to to cluster if $s^2 < p$ and furthermore it is information theoretically impossible to even estimate the model parameters from the graph when $s^2 < p$. Here we complete the proof of the conjecture by providing an efficient algorithm for clustering in a way that is correlated with the true partition when $s^2 > p$. A different independent proof of the same result was recently obtained by Laurent Massoulie.

math.PR↗

arXiv2015

Local Algorithms for Block Models with Side Information

There has been a recent interest in understanding the power of local algorithms for optimization and inference problems on sparse graphs. Gamarnik and Sudan (2014) showed that local algorithms are weaker than global algorithms for finding large independent sets in sparse random regular graphs. Montanari (2015) showed that local algorithms are suboptimal for finding a community with high connectivity in the sparse Erdős-Rényi random graphs. For the symmetric planted partition problem (also named community detection for the block models) on sparse graphs, a simple observation is that local algorithms cannot have non-trivial performance. In this work we consider the effect of side information on local algorithms for community detection under the binary symmetric stochastic block model. In the block model with side information each of the $n$ vertices is labeled $+$ or $-$ independently and uniformly at random; each pair of vertices is connected independently with probability $a/n$ if both of them have the same label or $b/n$ otherwise. The goal is to estimate the underlying vertex labeling given 1) the graph structure and 2) side information in the form of a vertex labeling positively correlated with the true one. Assuming that the ratio between in and out degree $a/b$ is $Θ(1)$ and the average degree $ (a+b) / 2 = n^{o(1)}$, we characterize three different regimes under which a local algorithm, namely, belief propagation run on the local neighborhoods, maximizes the expected fraction of vertices labeled correctly. Thus, in contrast to the case of symmetric block models without side information, we show that local algorithms can achieve optimal performance for the block model with side information.

stat.ML↗

arXiv2015

Quickest Online Selection of an Increasing Subsequence of Specified Size

Given a sequence of independent random variables with a common continuous distribution, we consider the online decision problem where one seeks to minimize the expected value of the time that is needed to complete the selection of a monotone increasing subsequence of a prespecified length $n$. This problem is dual to some online decision problems that have been considered earlier, and this dual problem has some notable advantages. In particular, the recursions and equations of optimality lead with relative ease to asymptotic formulas for mean and variance of the minimal selection time.

math.PR↗

arXiv2015

Strong Contraction and Influences in Tail Spaces

We study contraction under a Markov semi-group and influence bounds for functions in $L^2$ tail spaces, i.e. functions all of whose low level Fourier coefficients vanish. It is natural to expect that certain analytic inequalities are stronger for such functions than for general functions in $L^2$. In the positive direction we prove an $L^{p}$ Poincaré inequality and moment decay estimates for mean $0$ functions and for all $1<p<\infty$, proving the degree one case of a conjecture of Mendel and Naor as well as the general degree case of the conjecture when restricted to Boolean functions. In the negative direction, we answer negatively two questions of Hatami and Kalai concerning extensions of the Kahn-Kalai-Linial and Harper Theorems to tail spaces. That is, we construct a function $f\colon\{-1,1\}^{n}\to\{-1,1\}$ whose Fourier coefficients vanish up to level $c \log n$, with all influences bounded by $C \log n/n$ for some constants $0<c,C< \infty$. We also construct a function $f\colon\{-1,1\}^{n}\to\{0,1\}$ with nonzero mean whose remaining Fourier coefficients vanish up to level $c' \log n$, with the sum of the influences bounded by $C'(\mathbb{E}f)\log(1/\mathbb{E}f)$ for some constants $0<c',C'<\infty$.

math.PR↗